Wolfgang Schwarz

Spelling out a Dutch Book argument

Dutch Book arguments are often used to justify various epistemic
norms – in particular, that credences should obey the
probability axioms and that they should evolve by
condionalization. Roughly speaking, the argument is that if someone
were to violate these norms, then they would be prepared to accept
bets which amount to a guaranteed loss, and that seems
irrational.

But it's hard to spell out how exactly the argument is meant to go. In
fact, I'm not aware of any satisfactory statement. Here's my
attempt.

For concreteness, I'll focus on the argument for probabilism,
but the case of conditionalization is similar.

The argument begins with an uncontroversial mathematical fact, the
Dutch Book Theorem:

Let a unit bet on a proposition A be a deal that pays $1 if A
is true and otherwise $0. Suppose for any proposition A, an agent is
prepared to buy a unit bet on A for up to $Cr(A) – the dollar
value corresponding to her credence in A – and she is prepared to sell
a unit bet on A for $Cr(A) or more. If her credences do not satisfy
the axioms of non-negativity, normalization, and finite additivity,
she will then be prepared to buy and/or sell unit bets in such a way that if she makes all these transactions she incurs
a guaranteed loss.

How do we get from here to an argument that rational credences
should conform to the probability axioms? A few problems immediately
stand out. (See Hajek
2008.)

First, the theorem seems to say nothing about people who aren't
prepared to trade bets in accordance with their expected monetary
payoff. Surely epistemic rationality does not require having a
utility function that is linear with respect to monetary payoff. An
epistemically rational agent need not care about money at all.

Worse, even if an agent does care only about money, and her utility
function is linear with respect to monetary payoff, she ought not to
be prepared to buy a unit bet on any proposition A for up to the
dollar value corresponding to her credence in A. For example, let A be
the proposition that the agent will not buy any bets today. An agent's
credence in A may well be high, yet she ought not to pay much for the
corresponding bet, since doing so would render A false.

Even if the relevant propositions are unaffected by the considered
bets, there can be interference effects between different bets. For
example, what if our agent has earlier bought a high-stakes bet on the
proposition that she will not buy any more bets today? Then she may
not be prepared to buy a unit bet on any proposition
whatsoever. Relatedly, the Dutch Book argument for finite additivity
involves at least three bets; if a probabilistically incoherent agent
cares about the net outcome of all her transactions, rather than
myopically about the isolated outcome of whatever transaction she
currently considers, it is not clear why she ought to make all three
transactions. (This is the "package principle
objection". Interestingly, it seems not to arise for the case of
conditionalisation, as Skyrms 1993 shows.)

Stepping back, why is the mere possibility of making a sure loss
normatively relevant? After all, as Lewis said, "there aren't so many
sneaky Dutchmen around".

Finally, why is the possibility of financial loss a sign of
epistemic, rather than practical irrationality?

A neat way to get around most of these problems, which I haven't
seen in the literature, is to invoke some broadly Humean principles
about the independence of belief and desire. In outline, the idea is
that for any probabilistically incoherent agent X there is a possible
agent Y who (1) has the same credences as X, (2) only cares about the
monetary payoff of whatever transaction they presently consider, and
(3) is offered the relevant bets that make up a Dutch Book. Y then
makes a sure (and avoidable) loss, despite trying to get as much money
as possible. Something has gone wrong. But the fault must lie in Y's
beliefs, for neither her utilities nor her decision process is
faulty. So Y's beliefs are irrational. But Y's beliefs are identical
to X's. So X's beliefs are irrational.

That's the outline. Let's fill in the details.

Let X be an arbitrary agent whose credences violate one of the
probability axioms. Our aim is to show that X is epistemically
irrational.

Let Y be a possible counterpart of X with the same (centred)
credences. But Y has strange desires. Whenever Y is offered a monetary
gamble, she only myopically cares about the net amount of money she
will make through the present transaction. Specifically, if Y has the
option to buy a unit bet on some proposition A for some amount $x,
then the only thing she cares about is whether she will eventually (i)
win $1 after having paid $x, or (ii) not win after having paid $x, or
(iii) not win after not having bought the bet; the utility she assigns
to these outcomes are, respectively, (i) $1-$x, (ii) -$x, (iii)
$0. Similarly, mutatis mutandis, if Y has the opportunity to sell a
unit bet.

I'll also stipulate that when faced with a choice, Y always chooses an
option with maximal expected utility.

All this still doesn't ensure that Y is prepared to pay up to
$Cr(A) for a unit bet on A because her credence in A may be affected
by getting an offer to buy the bet or even by the act of buying (as
when A is the proposition that she won't buy any bets today). If we
want Y to accept a Dutch Book that involves several transactions, we
must also ensure that, say, buying the first bet does not affect the
expected utility of buying the second.

I'm not sure how best to get around these problems. Here's a brute
force response.

Let's say that X's (and Y's) credence function is stable with
respect to some propositions A,B,...,N iff Y regards $Cr(A) as the
fair price for a unit bet on A, $Cr(B) as the fair price for a unit
bet on B conditional on having bought/sold a unit bet on A, and so
on. That is, a credence function is stable with respect to a list of
propositions if the credence in each proposition on the list is not
affected by whether a bet on that proposition or a proposition earlier
in the list has been bought or sold.

If we assume that there is a list of propositions for which
X's credences are stable and violate the probability axioms, we can stipulate that Y
is made the relevant offers and gets caught in a Dutch Book.

So we need to assume that X's probabilistic incoherence isn't
restricted to unstable parts of her credence function. To get a
general argument for probabilism, we'll need the following premise.

Premise 1. If any restriction of an agent's
credence function to stable propositions should satisfy the
probability axioms, then so should her entire credence
function.

(Here and throughout, a violation of the probability axioms means
that either (i) some proposition has negative probability, or (ii) the
tautology does not have probability 1, or (iii) there are disjoint
propositions whose disjunction has a probability that is not the sum
of the probability of the disjuncts. Boolean closure is not treated as
an axiom.)

The motivation for Premise 1 is that the probability axioms are
supposed to be general consistency requirements on rational
belief. They are meant to hold for beliefs or any kind, not just for
beliefs with a specific content.

Jeffrey makes a similar move in Subjective Probability: The Real
Thing (pp.4f.):

If the truth [of a proposition about distant planets] is
not known in my lifetime, I cannot cash the ticket even if it is
really a winner. But some probabilities are plausibly represented by
prices, e.g., probabilities of the hypotheses about athletic contests
and lotteries that people commonly bet on. And it is plausible to
think that the general laws of probability ought to be the same for
all hypotheses – about planets no less than about ball
games. If that is so, we can justify laws of probability if we can
prove all betting policies that violate them to be
inconsistent.

Jeffrey here assumes that there's a reasonably wide set of
propositions for which our credences match our betting prices. I
assume something much weaker. But I'd still like to know how to do better.

(One reassuring fact to keep in mind is that probabilistic
incoherence is infectious: if, for example, your credence in an
exclusive disjunction A v B is not the sum of your credences in A and
B, there will be lots of other propositions for which you'll violate
additivity. So it requires some fine-tuning to limit incoherence to
unstable fragments of a credence function.)

Moving on, recall that X was an arbitrary agent whose credences
fail to satisfy the probability axioms. We want to show that X is
epistemically irrational. By Premise 1, we can assume without loss of
generality that there are some propositions with respect to which X's
credence function is stable but fails to satisfy the probability
axioms.

The next premise is that all the differences between X and Y are
irrelevant to whether the credence function shared by X and Y is
epistemically rational. More precisely:

Premise 2. If X is epistemically rational, then so
is Y.

The basic idea is that whether someone's beliefs are epistemically
rational does not depend on her goals or desires. If we want to know
whether it is epistemically rational (as opposed to practically
useful) for an agent to have such-and-such beliefs, we don't need to
know anything about her goals or desires.

I've also assumed that Y is an expected utility maximizer, which X
may not be. But again, arguably the epistemic rationality of someone's
beliefs would not be undermined by finding that they are an expected
utility maximizer.

Finally, Premise 2 implies that it does not affect the epistemic
rationality of an agent's beliefs if they are about to be offered a
series of bets. (That's the final difference between X and Y.)

Premise 2 looks fairly good to me.

Now the Dutch Book theorem tells us that there are certain
transactions that Y is prepared to make that would amount to a
guaranteed loss. Let's stipulate that Y is made the relevant offers
and thus really does make a sure loss.

The next premise states that something has then gone wrong.

Premise 3. It is irrational of Y to make choices
that together amount to a sure loss (a loss she could have avoided by
making different choices).

Here the guiding idea is that it is irrational for an agent whose
sole aim is to maximize monetary profit to knowingly and avoidably
enter transactions that are logically guaranteed to cost her
money.

I'm not entirely happy with Premise 3. The problem is that, by
assumption, Y does not care about her net wealth. When offered a
series of choices, she only cares about the net outcome of the
present choice. It would be nicer if we could stipulate that Y
cares about the net payoff of all the choices she's about to
make. This would make Premise 3 quite compelling, I think. But then
we'd need to explain why Y accepts the individual deals that together
constitute a Dutch Book. (Skyrms offers such an explanation in his
1993 paper on conditionalization, but the argument sadly doesn't
generalize to the case of finite additivity.)

The rest of the argument is simple.

Premise 4. If an agent makes irrational choices,
then either she is epistemically irrational or her desires are irrational
or her acts don't maximize expected utility.

Premise 5. Y's desires are not irrational.

Y's myopic desires are admittedly weird, but on a suitably weak
notion of rationality, I think they should pass. Since Y maximizes
expected utility, we can conclude that Y's credences are
irrational. Intuitively, Y misjudges the profitability of the relevant
bets.

So Y is epistemically irrational. By Premise 2, it follows that X is
epistemically irrational. QED.

Comments

No comments yet.

Add a comment

Please leave these fields blank (spam trap):

Name: Email: (not displayed)

No HTML please.You can edit this comment until 30 minutes after posting.